alpha.math.uga.edualpha.math.uga.edu/~pete/triangle-110315.pdf · ALGEBRAIC CURVES UNIFORMIZED BY...

ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCESUBGROUPS OF TRIANGLE GROUPS

PETE L. CLARK AND JOHN VOIGHT

Abstract. We construct certain subgroups of hyperbolic triangle groups which we call“congruence” subgroups. These groups include the classical congruence subgroups of SL2(Z),Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimuracurves. We determine the field of moduli of the curves associated to these groups andthereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

Contents

1. Introduction 12. Triangle groups 73. Galois Bely̆ı maps 104. Fields of moduli 135. Congruence subgroups of triangle groups 156. Weak rigidity 217. Conjugacy classes, fields of rationality 238. Subgroups of PSL2(Fq) and PGL2(Fq) and weak rigidity 269. Proof of Theorems 3210. Examples 37References 46

1. Introduction

Motivation. The rich arithmetic and geometric theory of classical modular curves, quo-tients of the upper half-plane by subgroups of SL2(Z) defined by congruence conditions, hasfascinated mathematicians since at least the nineteenth century. One can see these curvesas special cases of several distinguished classes of curves. Fricke and Klein [25] investigatedcurves obtained as quotients by Fuchsian groups which arise from the unit group of certainquaternion algebras, now called arithmetic groups. Later, Hecke [32] investigated his trianglegroups, arising from reflections in the sides of a hyperbolic triangle with angles 0, π/2, π/nfor n ≥ 3. Then in the 1960s, amidst a flurry of activity studying the modular curves, Atkinand Swinnerton-Dyer [1] pioneered the study of noncongruence subgroups of SL2(Z). Inthis paper, we consider a further direction: we introduce a class of curves arising from cer-tain subgroups of hyperbolic triangle groups. These curves share many appealing propertiesin common with classical modular curves despite the fact that their uniformizing Fuchsiangroups are in general not arithmetic groups.

Date: November 3, 2015.1

To motivate the definition of this class of curves, we begin with the modular curves. Let pbe prime and let Γ(p) ⊆ PSL2(Z) = Γ(1) be the subgroup of matrices congruent to (plus orminus) the identity modulo p. Then Γ(p) acts on the completed upper half-planeH∗, and thequotient X(p) = Γ(p)\H∗ can be given the structure of Riemann surface, a modular curve.The subgroup G = Γ(1)/Γ(p) ⊆ Aut(X(p)) satisfies G ' PSL2(Fp) and the natural mapj : X(p)→ X(p)/G ' P1C is a Galois branched cover ramified at the points {0, 1728,∞}.

So we are led to study class of (smooth, projective) curves X over C with the propertythat there exists a subgroup G ⊆ Aut(X) with G ' PSL2(Fq) or G ' PGL2(Fq) for a primepower q such that the map X → X/G ' P1 is a Galois branched cover ramified at exactlythree points.

Bely̆ı [3, 4] proved that a curve X over C can be defined over the algebraic closure Q ofQ if and only if X admits a Bely̆ı map, a nonconstant morphism f : X → P1C unramifiedaway from 0, 1,∞. So, on the one hand, three-point branched covers are indeed ubiquitous.On the other hand, there are only finitely many curves X up to isomorphism of given genusg ≥ 2 which admit a Galois Bely̆ı map (Remark 3.10). We call a curve which admitsa Galois Bely̆ı map a Galois Bely̆ı curve. Galois Bely̆ı curves are also called quasiplatonicsurfaces [26, 85], owing to their connection with the Platonic solids, or curves with manyautomorphisms because they are equivalently characterized as the locus on the moduli spaceMg(C) of curves of genus g at which the function [C] 7→ # Aut(C) attains a strict localmaximum. For example, the Hurwitz curves, those curves X with maximal automorphismgroup # Aut(X) = 84(g − 1) for their genus g, are Galois Bely̆ı curves, as are the Fermatcurves xn + yn = zn for n ≥ 3. (Part of the beauty of this subject is that the same objectcan be viewed from many different perspectives, and the natural name for an object dependson this view.)

So Galois Bely̆ı curves with Galois group G = PSL2(Fq) and G = PGL2(Fq) generalizethe classical modular curves and bear further investigation. In this article, we study theexistence of these curves, and we then consider one of the most basic properties about them:the fields over which they are defined.

Existence. To state our first result concerning existence we use the following notation. Fors ∈ Z≥2, let ζs = exp(2πi/s) and λs = ζs + 1/ζs = 2 cos(2π/s); by convention we let ζ∞ = 1and λ∞ = 2.

Let a, b, c ∈ Z≥2 ∪ {∞} satisfy a ≤ b ≤ c. Then we have the following extension of fields:

(*)

F (a, b, c) = Q(λ2a, λ2b, λ2c)

E(a, b, c) = Q(λa, λb, λc, λ2aλ2bλ2c)

D(a, b, c) = Q(λa, λb, λc)

Q

We have E(a, b, c) ⊆ F (a, b, c) since λ22a = λa + 2 (the half-angle formula), and consequentlythis extension has degree at most 4 and exponent at most 2. Accordingly, a prime p of

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E(a, b, c) (by which we mean a nonzero prime ideal in the ring of integers of E(a, b, c)) thatis unramified in F (a, b, c) either splits completely or has inertial degree 2.

The triple (a, b, c) is hyperbolic if

χ(a, b, c) =1

a+

1

b+

1

c− 1 < 0.

Our first main result is as follows.

Theorem A. Let (a, b, c) be a hyperbolic triple with a, b, c ∈ Z≥2. Let p be a prime ofE(a, b, c) with residue field Fp and suppose p - 2abc. Then there exists a G-Galois Bely̆ı map

X(a, b, c; p)→ P1

with ramification indices (a, b, c), where

G =

{PSL2(Fp), if p splits completely in F (a, b, c);PGL2(Fp), otherwise.

We have stated Theorem A in a simpler form; for a more general statement, includingthe case when p | 2 or when one or more of a, b, c is equal to ∞, see Theorem 9.1. In somecircumstances (depending on a norm residue symbol), one can also take primes dividing abc(see Remark 5.23).

Theorem A generalizes work of Lang, Lim, and Tan [38] who treat the case of Hecketriangle groups using an explicit presentation of the group (see also Example 10.4), andwork of Marion [45] who treats the case a, b, c prime. This also complements celebrated workof Macbeath [41], providing an explicit way to distinguish between projective two-generatedsubgroups of PSL2(Fq) by a simple splitting criterion. Theorem A overlaps with work ofConder, Potočink, and Širáň [16] (they also give several other references to work of thiskind).

The construction of Galois Bely̆ı maps in Theorem A arises from another equivalent char-acterization of Galois Bely̆ı curves (of genus ≥ 2) as compact Riemann surfaces of the formΓ\H, where Γ is a finite index normal subgroup of the hyperbolic triangle group

∆(a, b, c) = 〈δ̄a, δ̄b, δ̄c | δ̄aa = δ̄bb = δ̄cc = δ̄aδ̄bδ̄c = 1〉 ⊂ PSL2(R)for some a, b, c ∈ Z≥2, where by convention we let δ̄∞∞ = 1. (See Sections 1–2 for more detail.The bars may seem heavy-handed here, but signs play a delicate and somewhat importantrole in the development, so we include this as part of the notation for emphasis.) Phrasedin this way, Theorem A asserts the existence of a normal subgroup

∆(p) = ∆(a, b, c; p) E ∆(a, b, c) = ∆

with quotient ∆/∆(p) = G as above. In a similar way, one obtains curves X0(a, b, c; p) byconsidering the quotient of X(a, b, c; p) by the subgroup of upper-triangular matrices—thesecurves are analogous to the classical modular curves X0(p) as quotients of X(p).

Arithmeticity. A Fuchsian group is arithmetic if it is commensurable with the group ofunits of reduced norm 1 of a maximal order in a quaternion algebra. A deep theorem ofMargulis [44] states that a Fuchsian group is arithmetic if and only if it is of infinite indexin its commensurator. By work of Takeuchi [77], only finitely many of the groups ∆(a, b, c)are arithmetic: in these cases, the curves X(a, b, c; p) are Shimura curves (arising from full

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congruence subgroups) and canonical models were studied by Shimura [62] and Deligne [19].Indeed, the curves X(2, 3,∞; p) are the classical modular curves X(p) and the Galois Bely̆ımap j : X(p)→ P1 is associated to the congruence subgroup Γ(p) ⊆ PSL2(Z). Several otherarithmetic families of Galois Bely̆ı curves have seen more detailed study, most notably thefamily X(2, 3, 7; p) of Hurwitz curves. (It is interesting to note that the arithmetic trianglegroups are among the examples given by Shimura [62, Example 3.18]!) Aside from thesefinitely many triples, the triangle group ∆ = ∆(a, b, c) is not arithmetic, and our results canbe seen as a generalization in this nonarithmetic context.

However, we still have an embedding inside an arithmetic group Γ, following Takeuchi[77] and later work of Tretkoff (née Cohen) and Wolfart [13]: our curves are obtained viapullback

∆(p)\H ↪ //

��

Γ(p)\Hs

��P1 = ∆\H ↪ // Γ(1)\Hs

from a branched cover of quaternionic Shimura varieties, and this promises further arith-metic applications. Accordingly, we call the subgroups ∆(a, b, c; p) we construct congruencesubgroups of ∆ in analogy with the classical case of modular curves, since they arise fromcertain congruence conditions on matrix entries, and we call the curves X(a, b, c; p) andtheir cousins X0(a, b, c; p) triangle modular curves. (In some contexts, the term congruenceis used only for arithmetic groups; we propose the above extension of this terminology tonon-arithmetic groups.) For a fuller discussion of the arithmetic cases of Theorem A, seeExample 10.3.

Field of definition. Our second main result studies fields of definition. The modular curveX(p), a Riemann surface defined over C, has a model as an algebraic curve defined over Q;we seek a similar (nice, explicit) result for our class of curves. For a curve X defined over C,the field of moduli M(X) of X is the fixed field of the group {σ ∈ Aut(C) : Xσ ' X}, whereXσ is the base change of X by the automorphism σ ∈ Aut(C). A field of definition for Xclearly contains the field of moduli of X, so if X has a minimal field of definition F then Fis necessarily equal to the field of moduli.

We will need two refinements of this notion. First, we define the notion for branchedcovers. We say that two Bely̆ı maps f : X → P1 and f ′ : X ′ → P1 are isomorphic (over C orQ) if there exists an isomorphism h : X ∼−→ X ′ that respects the branched covers, i.e., suchthat f = f ′ ◦h. We define the field of moduli M(X, f) of a Bely̆ı map analogously. A GaloisBely̆ı map can always be defined over its field of moduli (Lemma 4.1) as mere cover.

But we will also want to keep track of the Galois automorphisms of the branched cover.For a finite group G, a G-Galois Bely̆ı map is a Bely̆ı map f : X → P1 equipped with anisomorphism i : G

∼−→ Gal(f) between G and the Galois (monodromy) group of f , and anisomorphism of G-Galois Bely̆ı maps is an isomorphism h of Bely̆ı maps that identifies i withi′, i.e.,

h(i(g)x) = i′(g)h(x) for all g ∈ G and x ∈ X(C)4

so the diagram

X

i(g)��

h // X ′

i′(g)��

X

f

h // X ′

f ′~~P1

commutes. We define the field of moduli M(X, f,G) of a G-Galois Bely̆ı map f accordingly.For example, we have

M(X(p), j,PSL2(Fp)) = Q(√p∗) where p∗ = (−1)(p−1)/2p.

A G-Galois Bely̆ı map f can be defined over its field of moduli M(X, f,G) under the followingcondition: if G has trivial center Z(G) = {1} and G = Aut(X) (otherwise, take a furtherquotient).

On the one hand, we observe (Remark 4.7) that for any number field K there is a G-GaloisBely̆ı map f for some finite group G such that the field of moduli of (X, f,G) contains K.On the other hand, we will show that our curves have quite nice fields of definition. (Seealso work of Streit and Wolfart [72] who consider the case G ' Z/pZ o Z/qZ.)

We need one further bit of notation. For a prime p and integers a, b, c ∈ Z≥2, let Dp′(a, b, c)be the compositum of the fields Q(λs) with s ∈ {a, b, c} prime to p. (For example, if all ofa, b, c are divisible by p, then Dp′(a, b, c) = Q.) Similarly define Fp′(a, b, c).

Theorem B. Let X be a curve of genus g ≥ 2 and let f : X → P1 be a G-Galois Bely̆ı mapwith G ' PGL2(Fq) or G ' PSL2(Fq). Let (a, b, c) be the ramification indices of f .

Then the following statements hold.

(a) Let r be the order of Frobp in Gal(Fp′(a, b, c)/Q). Then

q =

{√pr, if G ' PGL2(Fq);

pr, if G ' PSL2(Fq).

(b) The map f as a mere cover is defined over its field of moduli M(X, f). Moreover,M(X, f) is an extension of Dp′(a, b, c)

〈Frobp〉 of degree d(X,f) ≤ 2. If a = 2 or q iseven, then d(X,f) = 1.

(c) The map f as a G-Galois Bely̆ı map is defined over its field of moduli M(X, f,G).Let

Dp′(a, b, c){√p∗} =

{Dp′(a, b, c)(

√p∗), if p | abc, pr is odd, and G ' PSL2(Fq);

Dp′(a, b, c) otherwise.

Then M(X, f,G) is an extension of Dp′(a, b, c){√p∗} of degree d(X,f,G) ≤ 2. If q is

even or p | abc or G ' PGL2(Fq), then d(X,f,G) = 1.

(Not all G-Galois Bely̆ı maps with G ' PGL2(Fq) or G ' PSL2(Fq) arise from theconstruction in Theorem A, but Theorem B applies to them all.)

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The various fields of moduli fit into the following diagram.

M(X,G, f)

d(X,f,G)≤2

M(X, f)

d(X,f)≤2

Dp′(a, b, c){√p∗} = Q(λa, λb, λc)p′{

√p∗}

Dp′(a, b, c)〈Frobp〉 = Q(λa, λb, λc)

〈Frobp〉p′

As a simple special case of Theorem B, we have the following corollary.

Corollary. Suppose f : X → P1 is a PSL2(Fq)-Galois Bely̆ı map with ramification indices(2, 3, c) and suppose p - 6c is prime and a primitive root modulo 2c. Then q = pr wherer = φ(2c)/2 and f is defined over Q. Moreover, the monodromy group Gal(f) is definedover an at most quadratic extension of Q(λp).

To prove Theorem B, we use a variant of the rigidity and rationality results which arise inthe study of the inverse Galois problem [43, 83] and apply them to the groups PSL2(Fq) andPGL2(Fq). We use the classification of subgroups of PSL2(Fq) generated by two elementsprovided by Macbeath [41]. The statements q =

√pr and q = pr, respectively, can be found

in earlier work of Langer and Rosenberger [39, Satz (4.2)]; our proof follows similar lines.Theorem B generalizes work of Schmidt and Smith [56, Section 3] who consider the case ofHecke triangle groups as well as work of Streit [70] and Džambić [20] who considers Hurwitzgroups, where (a, b, c) = (2, 3, 7).

Composite level. The congruence subgroups so defined naturally extend to compositeideals, and so they form a projective system (Proposition 9.7). For a prime p of E ande ≥ 1, let P (pe) be the group

P (pe) =

{PSL2(ZE/pe), if p splits completely in F ;PGL2(ZE/pe), otherwise

where ZE denotes the ring of integers of E. For an ideal n of ZE, let P (n) =∏

pe‖n P (pe), and

let P̂ = lim←−n P (n) be the projective limit of P (n) with respect to the ideals n with n - 6abc.

Theorem C. ∆(a, b, c) is dense in P̂ .

Kucharczyk [37] uses superstrong approximation for thin subgroups of arithmetic groupsto prove a version of Theorem C that shows that the closure of the image of ∆(a, b, c) is

an open subgroup of P̂ , in particular of finite index; our Theorem C is more refined, givingeffective control over the closure of the image.

Applications. The construction and analysis of these curves has several interesting appli-cations. Combining Theorems A and B, we see that the branched cover X(a, b, c; p) → P1realizes the group PSL2(Fp) or PGL2(Fp) regularly over the field M(X, f,G), a small exten-sion of a totally real abelian number field. (See Malle and Matzat [43], Serre [58, Chapters7–8], and Volklein [83] for more information and groups realized regularly by rigidity andother methods.)

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Moreover, the branched covers X(a, b, c; p)→ X(a, b, c) have applications in the Diophan-tine study of generalized Fermat equations. When c = ∞, Darmon [17] has constructed afamily of hypergeometric abelian varieties associated to the triangle group ∆(a, b, c). Theanalogous construction when c 6= ∞ we believe will likewise have important arithmetic ap-plications. (See also work of Tyszkowska [79], who studies the fixed points of a particularsymmetry of PSL2(Fp)-Galois Bely̆ı curves.)

Finally, it is natural to consider applications to the arithmetic theory of elliptic curves.Every elliptic curve E over Q is uniformized by a modular curve X0(N)→ E, and the theoryof Heegner points govern facets of the arithmetic of E: in particular, it controls the rank ofE(Q) when this rank is at most 1. By analogy, we are led to consider those elliptic curvesover a totally real field that are uniformized by a curve X(a, b, c; p)—there is some evidence[81] that the images of CM points generate subgroups of rank at least 2.

Organization. The paper is organized as follows. In Sections 2–4, we introduce trianglegroups, Bely̆ı maps, Galois Bely̆ı curves, and fields of moduli. In Section 5, we investigate indetail a construction of Takeuchi, later explored by Cohen and Wolfart, which realizes thecurves associated to triangle groups as subvarieties of quaternionic Shimura varieties, andfrom this modular embedding we define congruence subgroups of triangle groups. We nextintroduce in Section 6 the theory of weak rigidity which provides the statement of Galoisdescent we will employ. In Section 7, we set up the basic theory of PSL2(Fq), and in Section8 we recall Macbeath’s theory of two-generated subgroups of SL2(Fq). In Section 9, we putthe pieces together and prove Theorems A, B, and C. We conclude in Section 10 with severalexplicit examples.

The authors would like to thank Henri Darmon, Richard Foote, David Harbater, Hi-laf Hasson, Robert Kucharczyk, Jennifer Paulhus, Jeroen Sijsling, Jürgen Wolfart, and theanonymous referee for helpful discussions and comments, as well as Noam Elkies for his valu-able comments and encouragement. The second author was supported by an NSF CAREERAward (DMS-1151047).

2. Triangle groups

In this section, we review the basic theory of triangle groups. We refer to Magnus [42,Chapter II] and Ratcliffe [50, §7.2] for further reading.

Let a, b, c ∈ Z≥2 ∪ {∞} satisfy a ≤ b ≤ c. We say that the triple (a, b, c) is spherical,Euclidean, or hyperbolic according as the quantity

χ(a, b, c) =1

a+

1

b+

1

c− 1

is positive, zero, or negative. The spherical triples are (2, 3, 3), (2, 3, 4), (2, 3, 5), and (2, 2, c)with c ∈ Z≥2. The Euclidean triples are (2, 2,∞), (2, 4, 4), (2, 3, 6), and (3, 3, 3). All othertriples are hyperbolic.

We associate to a triple (a, b, c) the extended triangle group ∆ = ∆(a, b, c), the groupgenerated by elements −1, δa, δb, δc, with −1 central in ∆, subject to the relations (−1)2 = 1and

(2.1) δaa = δbb = δ

cc = δaδbδc = −1;

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by convention we let δ∞∞ = −1. We define the quotient

∆ = ∆(a, b, c) = ∆(a, b, c)/{±1}

and call ∆ a triangle group. We denote by δ̄ the image of δ ∈ ∆(a, b, c) in ∆(a, b, c).

Remark 2.2. Reordering generators permits our assumption that a ≤ b ≤ c without loss ofgenerality. Indeed, the defining condition δaδbδc = −1 is invariant under cyclic permutationsso ∆(a, b, c) ' ∆(b, c, a) ' ∆(c, a, b), and similarly the map which sends a generator toits inverse gives an isomorphism ∆(a, b, c) ' ∆(c, b, a). The same is true for the quotients∆(a, b, c).

The triangle groups ∆(a, b, c) with (a, b, c) earn their name from the following geometricinterpretation. Associated to ∆ is a triangle T with angles π/a, π/b, and π/c on the Riemannsphere, the Euclidean plane, or the hyperbolic plane according as the triple is spherical,Euclidean, or hyperbolic, where by convention we let 1/∞ = 0. (The case (a, b, c) = (2, 2,∞)is admittedly a bit weird; one must understand the term triangle generously in this case.)The group of isometries generated by reflections τa, τ b, τ c in the three sides of the triangleT is a discrete group with T itself as a fundamental domain. The subgroup of orientation-preserving isometries is generated by the elements δ̄a = τ bτ c, δ̄b = τ cτa, and δ̄c = τaτ b andthese elements generate a group isomorphic to ∆(a, b, c). A fundamental domain for ∆(a, b, c)is obtained by reflecting the triangle T in one of its sides. The sides of this fundamentaldomain are identified by the elements δ̄a, δ̄b, and δ̄c, and consequently the quotient space is aRiemann surface of genus zero. This surface is compact if and only if a, b, c 6=∞ (i.e., c 6=∞since a ≤ b ≤ c). We analogously classify the groups ∆(a, b, c) as spherical, Euclidean, orhyperbolic. We make the convention Z/∞Z = Z.

Example 2.3. For all a, b ≥ 2, ∆(a, b,∞) is canonically isomorphic to the free productZ/aZ ∗ Z/bZ. This group is Euclidean when a = b = 2 and otherwise hyperbolic.

(a) The group ∆(2, 2,∞) = Z/2Z ∗ Z/2Z can be geometrically realized as the groupof isometries of the Euclidean plane generated by reflections through two distinct,parallel lines. This yields the alternate presentation

∆(2, 2,∞) ' 〈σ, τ | σ2 = 1, στσ−1 = τ−1〉.

The group ∆(2, 2,∞) is sometimes called the infinite dihedral group.(b) We have ∆(2, 3,∞) ' Z/4Z ∗Z/2Z Z/6Z ' SL2(Z). It follows that ∆(2, 3,∞) =

Z/2Z ∗ Z/3Z ' PSL2(Z).(c) The group ∆(∞,∞,∞) = Z ∗ Z is free on two generators. We have ∆(∞,∞,∞) '

Ker(PSL2(Z)→ PSL2(Z/2Z)).(d) For n ∈ Z≥2, the groups ∆(2, n,∞) ' Z/2Z ∗ Z/nZ are called Hecke groups [32].

Example 2.4. The spherical triangle groups are all finite groups: we have ∆(2, 2, c) ' D2c,the dihedral group on 2c elements, and

∆(2, 3, 3) ' A4, ∆(2, 3, 4) ' S4, ∆(2, 3, 5) ' A5.

We have the exact sequence

(2.5) 1→ [∆,∆]→ ∆→ ∆ab → 18

where [∆,∆] denotes the commutator subgroup. If c 6=∞, then ∆ab = ∆/[∆,∆] is isomor-phic to the quotient of Z/aZ×Z/bZ by the cyclic subgroup generated by (c, c); when c =∞,we have ∆

ab ' Z/aZ × Z/bZ. Thus, the group ∆ is perfect (i.e. ∆ab = {1}) if and only ifa, b, c are relatively prime in pairs. We have [∆,∆] ' Z for (a, b, c) = (2,∞,∞), whereasfor the other Euclidean triples we have [∆,∆] ' Z2 [42, §II.4]. In particular, the Euclideantriangle groups are infinite and nonabelian, but solvable.

From now on, suppose (a, b, c) is hyperbolic. Then by the previous paragraph we can realize∆ = ∆(a, b, c) ↪→ PSL2(R) as a Fuchsian group, a discrete subgroup of orientation-preservingisometries of the upper half-plane H. Let H(∗) denote H together with the cusps of ∆(a, b, c):this is the number of instances of∞ among a, b, c. We write X(a, b, c) = ∆(a, b, c)\H(∗) ' P1Cfor the quotient space.

We lift this embedding to SL2(R) as follows. Suppose that b

Here H ≤n G (resp. H En G) means that H is an index n subgroup of G (resp. an index nnormal subgroup of G). Moreover, to avoid tedious proliferation of cases, we have in (2.9)removed our assumption that a ≤ b ≤ c. It follows from (2.8)–(2.9) that ∆(a, b, c) is maximalif and only if (a, b, c) is not of the form

(2.10) (a, b, b), (2, b, 2b), or (3, b, 3b)

with again a, b, c ∈ Z≥2 ∪ {∞} not necessarily in increasing order.A Fuchsian group Γ is arithmetic [6] if there exists a quaternion algebra B over a totally

real field F that is unramified at precisely one real place of F such that Γ is commensurablewith the image of the units of reduced norm 1 in an order O ⊆ B. Takeuchi [77, Theorem 3]has enumerated the arithmetic triangle groups ∆(a, b, c): there are 85 of them, falling into19 commensurability classes [78, Table (1)].

3. Galois Bely̆ı maps

In this section, we discuss Bely̆ı maps and Galois Bely̆ı curves and we relate these curvesto those uniformized by subgroups of triangle groups.

A branched cover of curves over a field k is a finite morphism of curves f : X → Y definedover k. A Bely̆ı map is a branched cover f : X → P1 over C which is unramified awayfrom {0, 1,∞}. An isomorphism of branched covers between f and f ′ is an isomorphismh : X

∼−→ X ′ that respects the covers, i.e., such that f = f ′ ◦ h.

Remark 3.1. Let f : X → P1 be a morphism of degree d > 1. By Riemann-Hurwitz, f isramified over at least two points of P1, and if f is ramified over exactly two points thenX ' P1. In the latter case, after identifying X with P1 we may adjust the target by a linearfractional transformation so as to have f(z) = zd.

A branched cover that is a Galois (with Galois group G), i.e. a covering whose correspond-ing extension of function fields is Galois, is called a Galois branched cover; if such a branchedcover is further equipped with an isomorphism i : G

∼−→ Gal(f) = Aut(X, f) ⊆ Aut(X),it is called a G-Galois branched cover. Note the distinction between the two! A curve Xthat possesses a Galois Bely̆ı map is called a Galois Bely̆ı curve. An isomorphism of G-Galoisbranched covers over k is an isomorphism h of branched covers that identifies i with i′, i.e.,

h(i(g)x) = i′(g)h(x) for all g ∈ G and x ∈ X(k)where k is an algebraic closure of k. (This distinction may seem irrelevant at first, but itis important if one wants to study properties not just the cover but also the Galois groupof the branched cover.) For a Galois branched cover f : X → P1, the ramification indexof P ∈ X(C) depends only on f(P ), so we record these indices as a triple (a1, . . . , an) ofintegers 1 < a1 ≤ · · · ≤ an and say that (a1, . . . , an) is the ramification type of f .

Remark 3.2. If X has genus at least 2 and X → X/G is a G-Galois Bely̆ı map, then thequotient X → X/Aut(X) is a Aut(X)-Galois Bely̆ı map.

Example 3.3. The map

f : P1 → P1

f(t) =t2(t+ 3)

4= 1 +

(t− 1)(t+ 2)2

410

is a Bely̆ı map, a branched cover ramified only over 0, 1,∞, with ramification indices (2, 2, 3).In particular, P1 is a Galois Bely̆ı curve. The Galois closure of f is a Galois Bely̆ı map P1 →P1 with Galois group S3 corresponding to the simplest spherical triangle group ∆(2, 2, 3): itis given by

f(t) =27t2(t− 1)2

4(t2 − t+ 1)3with f(t)− 1 = −(t− 2)

2(2t− 1)2(t+ 1)2

4(t2 − t+ 1)3.

It becomes an S3-Galois Bely̆ı map when it is equipped with the isomorphism

S3∼−→ Gal(f) ≤ Aut(P1) ' PGL2(C)

(1 2) 7→ (t 7→ 1− t)↔(−1 10 1

)(1 2 3) 7→

(t 7→ 1

1− t

)↔(

0 1−1 1

).

All examples of Galois Bely̆ı maps P1 → P1 arise in this way from the spherical trianglegroups, as in Example 2.4.

Example 3.4. We now consider Galois Bely̆ı maps E → P1 where E is a curve of genus 1over C. There is no loss in assuming that E has the structure of elliptic curve with neutralelement ∞ ∈ E(C). The elliptic curves with extra automorphisms present candidates forsuch maps.

The curve E : y2 = x3−x with j(E) = 1728 has G = Aut(E,∞) cyclic of order 4, and thequotient x2 : E → E/G ' P1 yields a Galois Bely̆ı map of degree 4 with ramification type(2, 4, 4) by a direct computation. This map as a G-Galois Bely̆ı map is minimally definedover Q(

√−1); the Belyi map itself is defined over Q.

Next we consider the curve with j(E) = 0 with G = Aut(E,∞) cyclic of order 6, fromwhich we obtain two Galois Bely̆ı maps. The first map is obtained by writing E : y2 = x3−1and taking the map x3 : E → E/G ' P1, a Galois Bely̆ı map of degree 6 with ramificationtype (2, 3, 6). The second is obtained by writing instead E : y2 − y = x3 (isomorphically)and the unique subgroup H < G of order 3, corresponding to the map y : E → E/H ' P1with ramification type (3, 3, 3). These maps are minimally defined over Q(

√−3) as Galois

Bely̆ı maps. Indeed, the inclusions (2.9) imply an inclusion ∆(3, 3, 3) E2 ∆(2, 3, 6), so theformer is the composition of the latter together with the squaring map.

One obtains further Galois Bely̆ı maps by precomposing these with an isogeny E → E.

Lemma 3.5. Up to isomorphism, the only Galois Bely̆ı maps E → P1 with E a genus 1curve over C are of the form

Eφ−→ E f−→ P1

where φ is an isogeny and f is one of the three Galois Bely̆ı maps in Example 3.4. Inparticular, the only Galois Bely̆ı curves E of genus 1 have j(E) = 0, 1728.

Proof. Let E → P1 be a Galois Bely̆ı map, where without loss of generality we may assumeE : y2 = f(x) is an elliptic curve in Weierstrass form with neutral element ∞. We claimthat j(E) = 0, 1728. We always have AutE = E(C) o Aut(E,∞). If G < AutE is a finitesubgroup, then G′ = G ∩ E(C) E G and G/G′ ⊆ Aut(E,∞), so E ′ = E/G′ is an ellipticcurve and E/G ' E/(G/G′). However, if j(E) 6= 0, 1728, then Aut(E,∞) = {±1}, so either

11

G = G′ and E/G is an elliptic curve, or G = ±G′ and the map E → E/G′ x−→ E/G ' P1 isramified at four points, the roots of f(x) and ∞. �

In view of Examples 3.3 and 3.4 and Lemma 3.5, from now on we may restrict our attentionto Galois Bely̆ı maps f : X → P1 with X of genus g ≥ 2. These curves can be characterizedin several equivalent ways.

Proposition 3.6 (Wolfart [85, 87]). Let X be a compact Riemann surface of genus g ≥ 2.Then the following are equivalent.

(i) X is a Galois Bely̆ı curve;(ii) The map X → X/Aut(X) is a Bely̆ı map;

(iii) There exists a finite index, torsion-free normal subgroup Γ E ∆(a, b, c) with a, b, c ∈Z≥2 and a complex uniformization Γ\H

∼−→ X; and(iv) There exists an open neighborhood U of [X] (with respect to the complex analytic

topology) in the moduli space Mg(C) of curves of genus g such that # Aut(X) ># Aut(Y ) for all [Y ] ∈ U \ {[X]}.

Remark 3.7. Proposition 3.6 implies that Riemann surfaces uniformized by subgroups ofnon-cocompact hyerperbolic triangle groups are also uniformized by subgroups of cocompacthyperbolic triangle groups. More precisely: let a′, b′ ∈ Z≥2 ∪ {∞}, (a′, b′) 6= (2, 2), and letΓ′ ⊂ ∆(a′, b′,∞) be a finite index subgroup (not necessarily torsionfree). Then Γ′\H(∗) →∆(a′, b′,∞)\H(∗) is a Galois Bely̆ı map, so by Proposition 3.6 there are a, b, c ∈ Z≥2 anda finite index, normal torsionfree subgroup Γ ⊂ ∆(a, b, c) such that Γ′\H(∗) ' Γ\H. Thecase of PSL2(Fq)-Galois Bely̆ı curves uniformized by subgroups of Hecke triangle groups istreated in detail by Schmidt and Smith [56, Prop. 4].

By the Riemann-Hurwitz formula, if X is a G-Galois Bely̆ı curve of type (a, b, c), then Xhas genus

(3.8) g(X) = 1 +#G

2

(1− 1

a− 1b− 1c

)= 1− #G

2χ(a, b, c).

Remark 3.9. The function of #G in (3.8) is maximized when (a, b, c) = (2, 3, 7). Combiningthis with Proposition 3.6(iv) we recover the Hurwitz bound

# Aut(X) ≤ 84(g(X)− 1).

Remark 3.10. There are only finitely many Galois Bely̆ı curves of any given genus g. By theHurwitz bound (3.9), we can bound #G given g ≥ 2, and for fixed g and #G there are onlyfinitely many triples (a, b, c) satisfying (3.8). Each ∆(a, b, c) is finitely generated so has onlyfinitely many subgroups of index #G. From this, one can extract an explicit upper bound;using a more refined approach, Schlage-Puchta and Wolfart [55, Theorem 1] showed that thenumber of isomorphism classes of Galois Bely̆ı curves of genus at most g grows like glog g.

Remark 3.11. Wolfart [87] gives a complete list of all Galois Bely̆ı curves of genus g = 2, 3, 4.Further examples of Galois Bely̆ı curves can be found in the work of Shabat and Voevodsky[59]. See Table 10.5 for the determination of all PSL2(Fq)-Galois Bely̆ı curves with genusg ≤ 24.

12

Example 3.12. Let f : X → P1 be a Bely̆ı map and let g : Y → P1 be its Galois closure.Then g is also a Bely̆ı map and hence Y is a Galois Bely̆ı curve. Note however that thegenus of Y may be much larger than the genus of X!

Condition Proposition 3.6(iii) leads us to consider curves arising from finite index normalsubgroups of the hyperbolic triangle groups ∆(a, b, c). If Γ ⊆ PSL2(R) is a Fuchsian group,write X(Γ) = Γ \H(∗). If X is a compact Riemann surface of genus g ≥ 2 with uniformizingsubgroup Γ ⊆ PSL2(R), so that X = X(Γ), then Aut(X) = N(Γ)/Γ, where N(Γ) is thenormalizer of Γ in PSL2(R). Moreover, the quotient X → X/Aut(X), obtained from themap X(Γ) → X(N(Γ)), is a Galois cover with Galois group Aut(X). By the results ofSection 1, if Γ ⊆ ∆(a, b, c) is a finite index normal subgroup then Aut(X(Γ)) is of the form∆′/Γ with an inclusion ∆ ⊆ ∆′ as in (2.8)–(2.9); if ∆ is maximal, then we have

(3.13) Aut(X(Γ)) ' ∆(a, b, c)/Γ.

4. Fields of moduli

In this section, we briefly review the theory of fields of moduli and fields of definition. SeeCoombes and Harbater [15] and Köck [36] for more detail.

The field of moduli M(X) of a curve X over C is the fixed field of the group

{σ ∈ Aut(X) : Xσ ' X}.In a similar way, we define the fields of moduli M(X, f) of a Bely̆ı map f : X → P1 andM(X, f,G) of a G-Galois Bely̆ı map.

Owing to a lack of rigidity, not every curve can be defined over its field of moduli. However,in our situation we have the following lemma.

Lemma 4.1. Let f : X → P1 be a Galois Bely̆ı map. Then f is defined over its field ofmoduli M(X, f).

More generally, let X be a Galois Bely̆ı curve with Galois Bely̆ı map f : X → X/Aut(X) 'P1 such that the associated triangle group ∆ is maximal (not of the form (2.10)). ThenM(X, f) = M(X) and X is defined over its field of moduli M(X).

Proof. Dèbes and Emsalem [18, §1] remark that the first statement follows from results ofCoombes and Harbater [15]. The proof was written down by Köck [36, Theorem 2.2].

The subtlety in the second statement is that a Bely̆ı map f is rigidified so that an automor-phism of f is required to act as the identity on P1. If one allows automorphisms of P1, thenthere may be additional descent of f and X, and in particular the quotient X → X/Aut(X)may be a branched cover of a genus zero curve ramified above at most three points but wouldthen not be a Bely̆ı map, according to our definition. One always has M(X, g) = M(X)where g : X → X/Aut(X) = V , by Dèbes and Emsalem [18, Theorem 3.1]: indeed, anyautomorphism σ(X) → X with σ ∈ Gal(Q/Q) induces an isomorphism of automorphismgroups and hence of the pair (X, g). (Or, the field of M(X) is the intersection of all fields ofdefinition of X, but over any field K where X is defined, so is g (since Aut(X) as a schemeis defined over K), so M(X, g) = M(X).)

So to conclude the second statement of the lemma, we will show that the map g is a Bely̆ımap according to our definition, and for that it suffices to show that each ramification pointon the target curve V is M(X)-rational (so in particular V 'M(X) P1). To do this, we note

13

that since the associated triangle group ∆(a, b, c) is maximal, by (2.10) the three indicesa, b, c are distinct; any automorphism of g preserves ramification indices and thus necessarilyfixes these ramification points, so the base descends with these three points marked and soin the canonical model of Dèbes and Emsalem they are defined over M(X, g) = M(X). �

Remark 4.2. The subtlety in Lemma 4.1 is noted by Streit and Wolfart [72, Theorem 1,Remark 1], and they discuss the possible misinterpretation of the proof in Wolfart [88,Theorem 5] and the subtlety in rigidifying the base curve. Girondo, Torres-Teigell, andWolfart [27, Lemma 1, Remark 1, Lemma 2] also give a proof of the second statement anddiscuss descent for certain non-maximal triangle groups.

Keeping track of the action of the automorphism group, we also have the following result.

Lemma 4.3. Let f : X → P1 be a G-Galois Bely̆ı map. Suppose that CAut(X)(G) = {1},i.e., the centralizer of G in Aut(X) is trivial. Then f and the action of Gal(f) ' G can bedefined over its field of moduli M(X, f,G).

Proof. By definition, an automorphism of f as a G-Galois Bely̆ı map is given by h ∈ Aut(X)such that hi(g)h−1 = i(g) for all g ∈ G, so under the hypothesis of the lemma, f has noautomorphisms. Thus f and Gal(f) can be defined over M(X, f,G) by the criterion of Weildescent. �

Remark 4.4. Let X be a curve which can be defined over its field of moduli F = M(X).Then the set of F -isomorphism classes of models for X over F is in bijection with the Galoiscohomology set H1(Gal(F/F ),Aut(X)), where Aut(X) is equipped with the natural actionof the absolute Galois group Gal(F/F ). Similar statements are true more generally for theother objects considered here, including Bely̆ı maps and G-Galois Bely̆ı maps.

As a consequence of Lemma 4.3, if G ' Aut(X) and G has trivial center Z(G) = {1},then f as a G-Galois Bely̆ı map can be defined over M(X, f,G). Under this hypothesis, ifK = M(X, f,G), then by definition the group G occurs as a Galois group over K(t), andin particular applying Hilbert’s irreducibility theorem [58, Chapter 3] we find that G occursinfinitely often as a Galois group over K.

Example 4.5. Let p be prime and let X(p)/C = Γ(p)\H∗ be the classical modular curve,parametrizing (generalized) elliptic curves E equipped with a basis of E[p] which is symplec-tic with respect to the Weil pairing. Then Aut(X(p)) ⊇ G = PSL2(Fp), and the quotientmap j : X → X/G ' P1, corresponding to the inclusion Γ(p) ⊆ PSL2(Z), is ramified overj = 0, 1728,∞ with indices 2, 3, p, so X(p) is a Galois Bely̆ı curve.

For p ≤ 5, the curve X(p) has genus 0 and thus AutX(p) = PGL2(C). For p ≥ 7, thecurve X(p) has genus at least three (the curve X(7) has genus 3 and is considered in moredetail in the following example), so AutX(p) is a finite group containing PSL2(Fp). In factwe have AutX(p) = PSL2(Fp), as was shown by Mazur, following Serre [46, p. 255]. Laterwe will recover this fact as a special case of a more general result.

The field of moduli of j : X → P1 is Q, and indeed this map (and hence X) admitsa canonical model over Q [35]. This model is not unique, since the set H1(Q,Aut(X)) isinfinite: in fact, every isomorphism class of Galois modules E[p] with E an elliptic curvegives a different element in this set.

For p > 2, let p∗ = (−1)(p−1)/2 (so Q(√p∗) is the unique quadratic subfield of Q(ζp)).

The field of moduli of the PSL2(Fp)-Galois Bely̆ı map j is Q(√p∗) when p > 2 and Q when

14

p = 2, and in each case the field of moduli is a field of definition [60, pp. 108-109]. Indeed,this follows from Weil descent when p ≥ 7 and can be seen directly when p = 2, 3, 5 as thesecorrespond to spherical triples (2, 3, p) (cf. Example 3.3).

Example 4.6. The Klein quartic curve [23]

X3Y + Y 3Z + Z3X = 0

has field of definition equal to its field of moduli, which is Q, and all elements of Aut(X) canbe defined over Q(

√−7) = Q(

√7∗). Although the Klein quartic is isomorphic to X(7) over

Q, as remarked by Livné, the Katz-Mazur canonical model of X(7) agrees with the Kleinquartic only over Q(

√−3). The issue here concerns the fields of definition of the special

points giving rise to the canonical model. We do not go further into this issue here, but formore on this in the case of genus 1, see work of Sijsling [67].

Remark 4.7. We consider again Remark 3.12. If the field of moduli of a Bely̆ı map f : X → P1is F then the field of moduli of its Galois closure g : Y → P1 as a Bely̆ı map contains F .Consequently, let F be a number field and let X be an elliptic curve such that Q(j(X)) = F .Then X admits a Bely̆ı map defined over F . The Galois closure g : Y → P1 therefore hasfield of moduli containing F , and so for any number field F , there exists a G-Galois Bely̆ımap such that any field of definition of this map contains F . Note that from Lemma 3.5that outside of a handful of cases, the associated Galois Bely̆ı curve Y has genus g(X) ≥ 2.This shows that Gal(Q/Q) acts faithfully on the set of isomorphism classes of G-Galois Bely̆ıcurves. However if X → P1 is a G-Galois Bely̆ı map and H ≤ G is a subgroup, then thefield of moduli of X → X/H can be smaller than the M(X, f,G).

Nevertheless, González-Diez and Jaikin-Zapirain [28] have recently shown that Gal(Q/Q)acts faithfully on the set of Galois Bely̆ı curves.

In view of Remark 4.7, we restrict our attention from the general setup to the special classof G-Galois Bely̆ı curves X where G = PSL2(Fq) or PGL2(Fq).

5. Congruence subgroups of triangle groups

In this section, we associate a quaternion algebra over a totally real field to a trianglegroup following Takeuchi [76]. This idea was also pursued by Cohen and Wolfart [13] withan eye toward results in transcendence theory, and further elaborated by Cohen, Itzykson,and Wolfart [11]. Here, we use this embedding to construct congruence subgroups of ∆. Werefer to Vignéras [80] for the facts we will use about quaternion algebras and Katok [34] asa reference on Fuchsian groups.

Let Γ ⊆ SL2(R) be a subgroup such that Γ = Γ/{±1} ⊆ PSL2(R) has finite coarea, so inparticular is Γ is finitely generated. Let

F = Q(tr Γ) = Q(tr γ)γ∈Γbe the trace field of Γ. Then F is a finitely generated extension of Q.

Suppose further that F is a number field, so F has finite degree over Q, and let ZF beits ring of integers. Let F [Γ] be the F -vector space generated by Γ in M2(R), and let ZF [Γ]denote the ZF -submodule of F [Γ] generated by Γ. By work of Takeuchi [75, Propositions2–3], the ring F [Γ] is a quaternion algebra over F . If further tr(Γ) ⊆ ZF , then ZF [Γ] is anorder in F [Γ].

15

Remark 5.1. Schaller and Wolfart [54] call a Fuchsian group Γ semi-arithmetic if its tracefield F = Q(tr Γ) is a totally real number field and {tr γ2 : γ ∈ Γ} is contained in the ring ofintegers of F . They ask if all semi-arithmetic groups are either arithmetic or subgroups oftriangle groups; this conjecture remains open. This is implied by a conjecture of Chudnovskyand Chudnovsky [10, Section 7]. The Chudnovskys’ conjecture is false if the group is notcocompact—this is implicit in work of McMullen and made explicit in work of Bouw andMöller [7, 8]—but may still be true in the compact case. See also work of Ricker [52].

Let (a, b, c) be a hyperbolic triple with 2 ≤ a ≤ b ≤ c ≤ ∞. As in section 2, associ-ated to the triple (a, b, c) is the triangle group ∆(a, b, c) ⊆ SL2(R) with ∆(a, b, c)/{±1} '∆(a, b, c) ⊆ PSL2(R). Let F = Q(tr ∆(a, b, c)) be the trace field of ∆(a, b, c). The generatingelements δs ∈ ∆(a, b, c) for s = a, b, c satisfy the quadratic equations

δ2s − λ2sδs + 1 = 0in B where λ2s is defined in (2.6).

Lemma 5.2 ([77, Lemma 2]). Let Γ ⊆ SL2(R). If γ1, . . . , γr generate Γ, then Q(tr Γ) isgenerated by tr(γi1 · · · γis) for {i1, . . . , is} ⊆ {1, . . . , r}.

By Lemma 5.2, we deduce

F = Q(tr ∆(a, b, c)) = Q(λ2a, λ2b, λ2c).

Taking traces in the equation

δaδb = −δ−1c = δc − λ2c,yields

− tr(δ−1c ) = −λ2c = tr(δaδb) = δaδb + (λ2b − δb)(λ2a − δa).Also we have

(5.3) δaδb + δbδa = λ2bδa + λ2aδb − λ2c − λ2aλ2b.Together with the cyclic permutations of these equations, we conclude that the elements1, δa, δb, δc form a ZF -basis for the order O = ZF [∆] ⊆ B = F [∆] (see also Takeuchi [77,Proposition 3]).

Lemma 5.4. The reduced discriminant of O is a principal ZF -ideal generated byβ = λ22a + λ

22b + λ

22c + λ2aλ2bλ2c − 4 = λa + λb + λc + λ2aλ2bλ2c + 2.

Proof. Let d be the discriminant of O. Then we calculate from the definition that

d2 = det

2 λ2a λ2b λ2cλ2a λ

22a − 2 −λ2c −λ2b

λ2b −λ2c λ22b − 2 −λ2aλ2c −λ2b −λ2a λ22c − 2

ZF = β2ZF .Alternatively, we compute a generator for d using the scalar triple product and (5.3) as

tr([δa, δb]δc) = tr((δaδb − δbδa)δc) = tr(2δaδb − (λ2bδa + λ2aδb − λ2c − λ2aλ2b)δc)= −4− λ2b tr(δaδc)− λ2a tr(δbδc) + λ22c + λ2aλ2bλ2c = β

since δaδc = −δ−1b and δbδc = −δ−1a . �16

Lemma 5.5. If P is a prime of ZF with P - 2abc, then P - β. If further (a, b, c) is not ofthe form (mk,m(k + 1),mk(k + 1)) with k,m ∈ Z, then P - β for all P - abc.

Proof. Let P be a prime of F such that P - abc. We have the following identity in the fieldQ(ζ2a, ζ2b, ζ2c) = K:

(5.6) β =

(ζ2bζ2cζ2a

+ 1

)(ζ2aζ2cζ2b

+ 1

)(ζ2aζ2bζ2c

+ 1

)(1

ζ2aζ2bζ2c+ 1

).

Let PK be a prime above P in K and suppose that PK | β. Then PK divides one of thefactors in (5.6).

First, suppose that PK | (ζ2bζ2cζ−12a + 1), i.e., we have ζ2bζ2c ≡ −ζ2a (mod PK). Supposethat PK - 2abc. Then the map (Z×K)tors → F

×PK

is injective. Hence ζ2bζ2c = −ζ2a ∈ K. Butthen embedding K ↪→ C by ζs 7→ e2πi/s in the usual way, this equality would then read

(5.7)1

b+

1

c= 1 +

1

a∈ Q/2Z.

However, we have

0 ≤ 1b

+1

c≤ 1 < 1 + 1

a< 2

for any a, b, c ∈ Z≥2 ∪ {∞} when a 6= ∞, a contradiction, and when a = ∞ we haveb = c =∞ which again contradicts (5.7).

Now suppose PK | 2 but still PK - abc. Then ker((Z×K)tors → F×P) = {±1}, so instead

we have the equation ζ2bζ2c = ±ζ2a ∈ K. Arguing as above, it is enough to consider theequation with the +-sign, which is equivalent to

1

b+

1

c=

1

a.

Looking at this equation under a common denominator we find that b | c, say c = kb.Substituting this back in we find that (k + 1) | b so b = m(k + 1) and hence a = km andc = mk(k + 1), and in this case we indeed have equality.

The case where PK divides the middle two factors is similar. The case where PK dividesthe final factor follows from the impossibility of

0 = 1 +1

a+

1

b+

1

c∈ Q/2Z

since (a, b, c) is hyperbolic. �

We have by definition an embedding

∆ ↪→ O×1 = {γ ∈ O : nrd(γ) = 1}(where nrd denotes the reduced norm) and hence an embedding

(5.8) ∆ = ∆/{±1} ↪→ O×1 /{±1}.In fact, the image of this map arises from a quaternion algebra over a smaller field, as

follows. Let ∆(2) denote the subgroup of ∆ generated by −1 and γ2 for γ ∈ ∆. Then ∆(2)is a normal subgroup of ∆, and the quotient ∆/∆(2) is an elementary abelian 2-group. Wehave an embedding

∆(2)/{±1} ↪→ ∆/{±1} = ∆.17

Recall the exact sequence (2.5):

1→ [∆,∆]→ ∆→ ∆ab → 1.

Here, ∆ab

is the quotient of Z/aZ × Z/bZ × Z/cZ by the subgroup (1, 1, 1). We obtain∆(2) ⊇ [∆,∆] as the kernel of the (further) maximal elementary 2-quotient of ∆ab. It followsthat the quotient ∆/∆(2) is generated by the elements δs for s ∈ {a, b, c} such that eithers =∞ or s is even, and

(5.9) ∆/∆(2) '

{0}, if at least two of a, b, c are odd;Z/2Z, if exactly one of a, b, c is odd;(Z/2Z)2, if all of a, b, c are even or ∞.

(See also Takeuchi [77, Proposition 5].)Consequently, ∆(2) is the normal closure of the set {−1, δ2a, δ2b , δ2c} in ∆. A modification

of the proof of Lemma 5.2 shows that the trace field of ∆(2) can be computed on thesegenerators (trace is invariant under conjugation). We have

tr δ2s = tr(λ2sδs − 1) = λ22s − 2 = λs − 2for s ∈ {a, b, c} and similarly

tr(δ2aδ2b ) = tr((λ2aδa − 1)(λ2bδb − 1)) = λ2aλ2bλ2c − λ22b − λ22a + 2

and

tr(δ2aδ2b δ

2c ) = tr((λ2aδa − 1)(λ2bδb − 1)(λ2cδc − 1)) = λ22a + λ22b + λ22c + λ2aλ2bλ2c − 2;

from these we conclude that the trace field of ∆(2) is equal to

(5.10) E = F (a, b, c) = Q(λ22a, λ22b, λ22c, λ2aλ2bλ2c) = Q(λa, λb, λc, λ2aλ2bλ2c).(See also Takeuchi [77, Propositions 4–5].)

Example 5.11. The Hecke triangle groups ∆(2, n,∞) for n ≥ 3 have trace field F = Q(λ2n)whereas the corresponding groups ∆(2) have trace field E = Q(λn), which is strictly containedin F if and only if n is even.

Let Λ = ZE[∆(2)] ⊆ A = E[∆(2)] be the order and quaternion algebra associated to ∆(2).By construction we have

(5.12) ∆(2)/{±1} ↪→ Λ×1 /{±1}.We then have the following fundamental result.

Proposition 5.13. The image of the natural homomorphism

∆ ↪→ O×1

{±1}↪→ NB(O)

F×

lies in the subgroup NA(Λ×)/E× via

(5.14)

∆ ↪→ NA(Λ)E×

↪→ NB(O)F×

δ̄s 7→ δ2s + 1, if s 6= 2;δ̄a 7→ (δ2b + 1)(δ2c + 1), if a = 2.

18

where s = a, b, c and N denotes the normalizer. The map (5.14) extends the natural embed-ding (5.12).

Proof of Proposition 5.13. First, suppose a 6= 2 (whence b, c 6= 2, by the assumption thata ≤ b ≤ c). In B, for each s = a, b, c, we have

(5.15) δ2s + 1 = λ2sδs;

since s 6= 2, so that λ2s 6= 0, this implies that δ2s + 1 has order s in A×/E× ⊆ B×/F× and

(δ2a + 1)(δ2b + 1)(δ

2c + 1) = λ2aλ2bλ2cδaδbδc = −λ2aλ2bλ2c ∈ E×,

so (5.14) defines a group homomorphism ∆ ↪→ A×/E×. The image lies in the normalizerNA(Λ) because ∆

(2) generates Λ and ∆ normalizes ∆(2). Finally, we have

(δ2s + 1)2 = λ22sδ

2s ∈ A,

so the map extends the natural embedding of ∆(2)/{±1}.If a = 2, the same argument applies, with instead

δ̄a 7→ (δ2b + 1)(δ2c + 1)

since (δ2b + 1)(δ2c + 1) = λ2bλ2c(−δ−1a ) = λ2bλ2cδa now has order 2 in A×/E

×, and necessarily

b, c > 2 since the triple is hyperbolic. �

Example 5.16. The triangle group ∆(2, 4, 6) has trace field F = Q(√

2,√

3). However, thegroup ∆(2, 4, 6)(2) has trace field E = Q and indeed we find an embedding ∆(2, 4, 6) ↪→NA(Λ)/Q× where Λ is a maximal order in a quaternion algebra A of discriminant 6 over Q.

Corollary 5.17. The following statements hold.

(a) We have Λ⊗ZE ZF ⊆ O.(b) If a 6= 2, the quotient O/(Λ⊗ZE ZF ) is annihilated by λ2aλ2bλ2c.(c) If a = 2, the quotient O/(Λ⊗ZE ZF ) is annihilated by λ2bλ2c.

Proof. This follows from (5.15) since a basis for O is given by 1, δa, δb, δc. �

We now define congruence subgroups of triangle groups. Let N be an ideal of ZF suchthat N is coprime to abc and either N is coprime to 2 or

(a, b, c) 6= (mk,m(k + 1),mk(k + 1)) with m, k ∈ Z.Then by Lemma 5.5, we have an isomorphism

(5.18) O ⊗ZF ZF,N ' M2(ZF,N)

where ZF,N denotes the N-adic completion of the ring ZF : this is the product of the com-pletions at P for P | N and thus is a finite product of discrete valuation rings. Any twomaximal orders in a split quaternion algebra over a discrete valuation ring R with fractionfield K are conjugate by an element of M2(K) [80, Théorème II.2.3], and it follows easily thatthe isomorphism (5.18) is unique up to conjugation by an element of GL2(ZF,N). Alternately(and perhaps more fundamentally) since the ring ZF,N has trivial Picard group, the resultfollows from a generalization of the Noether–Skolem Theorem [53, Corollary 12].

Let

(5.19) O(N) = {γ ∈ O : γ ≡ 1 (mod NO)}.19

The definition of O(N) does not depend on the choice of isomorphism in (5.18). Then O(N)×1is normal in O×1 and we have an exact sequence

1→ O(N)×1 → O×1 /{±1} → PSL2(ZF/N)→ 1

where surjectivity follows from strong approximation [80, Théorème III.4.3]. Let

∆(N) = ∆ ∩ O(N)×1 .

Then we have

(5.20)∆

∆(N)↪→ O

×1 /{±1}O(N)×1

' PSL2(ZF/N).

We conclude by considering the image of the embedding (5.20). Let n be the prime ofE = F (a, b, c) below N. Then n is coprime to the discriminant of Λ since the latter di-vides (λ2aλ2bλ2c)β by Corollary 5.17. Therefore, we may define Λ(n) analogously. Then byProposition 5.13, we have an embedding

(5.21) ∆ ↪→ NA(Λ)E×

↪→ A×

E×↪→ A

×n

E×n' PGL2(En)

where En denotes the completion of E at n. The image of ∆ in this map lies in PGL2(ZE,n)by (5.15) since λ2s ∈ Z×E,n for s = a, b, c (since n is coprime to abc). Reducing the image in(5.21) modulo n, we obtain a map

∆→ PGL2(ZE/n).

This map is compatible with the map ∆→ PSL2(ZF/N) inside PGL2(ZF/N), obtained bycomparing the images in the reduction modulo N of B×/F×, by Proposition 5.13.

We summarize the main result of this section.

Proposition 5.22. Let a, b, c ∈ Z≥2 ∪ {∞}. Let N be an ideal of ZF with N prime to abcand such that either N is prime to 2 or (a, b, c) 6= (mk,m(k + 1),mk(k + 1)) with m, k ∈ Z.Let n = ZE ∩N. Then there exists a homomorphism

φ : ∆(a, b, c)→ PSL2(ZF/N)

such that trφ(δ̄s) ≡ ±λ2s (mod N) for s = a, b, c. The image of φ lies in the subgroup

PGL2(ZE/n) ∩ PSL2(ZF/N) ⊆ PGL2(ZF/N).

Remark 5.23. We conclude this section with some remarks extending the primes P of F(equivalently, primes p of E) for which the construction applies.

First, we note that whenever P - β, the order O is maximal at P.Second, even for a ramified prime P (or p), we still can consider the natural map to

the completion; however, instead of PGL2(FP) we instead obtain the units of an order in adivision algebra over FP, a prosolvable group. Our interest remains in the groups PSL2 andPGL2, but this case also bears further investigation: see Takei [74] for some results in thecase where b = c =∞.

Third, we claim that

B '(λ22s − 4, β

F

)20

for any s ∈ {a, b, c}. Indeed, given the basis 1, δa, δb, δc, we construct an orthogonal basis forB as

1, 2δa − λ2a, (λ22a − 4)δb + (λ2aλ2b + 2λ2c)δa − (λ22aλ2b − λ2aλ2c + 2λ2b)

which gives rise to the presentation B '(

4− λ22a, βF

). The others follow by symmetry. It

follows that a prime P of ZF ramifies in B if and only if we have for the Hilbert symbol(quadratic norm residue symbol) (λ22s − 4, β)P = −1 for (any) s ∈ {a, b, c}. For example, if(a, b, c) = (2, 3, c) (with c ≥ 7), one can show that the quaternion algebra B is ramified atno finite place.

A similar argument [78, Proposition 2] shows that

A '(λ22b(λ

22b − 4), λ22bλ22cβ

E

).

For any prime p of E which is unramified in A, we can repeat the above construction, andwe obtain a homomorphism φ as in (5.22); the image can be analyzed by considering theisomorphism class of the local order Λp, measured in part by the divisibility of β by p.

6. Weak rigidity

In this section, we investigate some weak forms of rigidity and rationality for Galois coversof P1. We refer to work of Coombes and Harbater [15], Malle and Matzat [43], Serre [58,Chapters 7–8], and Volklein [83] for references. Our main result concerns three-point covers,but we begin by briefly considering more general covers.

Let G be a finite group. An n-tuple for G is a finite sequence g = (g1, . . . , gn) of elementsof G such that g1 · · · gn = 1. In our applications we will take n = 3, so we will not emphasizethe dependence on n, and refer to tuples. A tuple is generating if 〈g1, . . . , gn〉 = G. LetC = (C1, . . . , Cn) be a finite sequence of conjugacy classes of G. Let Σ(C) be the set ofgenerating tuples g = (g1, . . . , gn) such that gi ∈ Ci for all i.

The group Inn(G) = G/Z(G) of inner automorphims of G acts on Gn via

x · g = x · (g1, . . . , gn) = gx = (xg1x−1, . . . , xgnx−1)

and restricts to an action of Inn(G) on C.Suppose that G has trivial center, so Inn(G) = G. To avoid trivialities, suppose also that

Σ(C) 6= ∅. Then the action of Inn(G) on Σ(C) has no fixed points: if z ∈ G fixes g, then zcommutes with each gi hence with 〈g1, . . . , gn〉 = G, so z ∈ Z(G) = {1}.

Now suppose that n = 3; we call a 3-tuple a triple. For every generating triple g, we obtainfrom the Riemann Existence Theorem [83, Theorem 2.13] a G-Galois branched coveringX(g) → P1 defined over Q with ramification type g over 0, 1,∞ and Galois group G. Twosuch covers f : X(g) → P1 and f ′ : X(g′) → P1 are isomorphic as covers if there existsan isomorphism h : X(g)

∼−→ X(g′) such that f = f ′ ◦ h; such an isomorphism from f tof ′ corresponds to an element ϕ ∈ Aut(G) such that ϕ(g) = (ϕ(g1), . . . , ϕ(gn)) = g′, andconversely.

We will have need also of a more rigid notion. A G-Galois branched cover is a branchedcover f : X → P1 equipped with an isomorphism i : G ∼−→ Aut(X, f). Two G-Galoisbranched covers (f, i) and (f ′, i′) are isomorphic (as G-Galois branched covers) if and only if

21

there is an isomorphism from f to f ′ that maps i to i′; such an isomorphism corresponds toan element x ∈ G such that

gx = (g′)x

and conversely.The group Gal(Q/Q) acts on the set of generating tuples for G up to automorphism

(or simply inner automorphism) via its action on the covers. Coming to grips with themysteries of this action in general is part of Grothendieck’s program of dessin d’enfants [30]:to understand Gal(Q/Q) via its faithful action on the fundamental group of P1Q \ {0, 1,∞}.There is one part of the action which is understood, coming from the maximal abelianextension of Q generated by roots of unity.

Let ζs = exp(2πi/s) ∈ C be a primitive sth root of unity for s ∈ Z≥2. The groupGal(Qab/Q) acts on tuples via the cyclotomic character χ: for σ ∈ Gal(Qab/Q) and a tripleg, we have σ ·g is uniformly conjugate to (gχ(σ)1 , . . . , g

χ(σ)n ) where if gi has order mi then g

χ(σ)i

is conjugate to gaii , where σ(ζmi) = ζaimi

. This action becomes an action on conjugacy classesin purely group theoretic language as follows. Let m be the exponent of G. Then the group(Z/mZ)× acts on G by s · g = gs for s ∈ (Z/mZ)× and g ∈ G and this induces an action onconjugacy classes. Pulling back by the canonical isomorphism Gal(Q(ζm)/Q)

∼−→ (Z/mZ)×defines the action of Gal(Q(ζm)/Q) and hence also Gal(Qab/Q) on the set of triples for G.

Let Hr ⊆ Gal(Q(ζm)/Q) be the kernel of this action:

Hr = {s ∈ (Z/mZ)× : Cs = C for all conjugacy classes C}.

The fixed field F r(G) = Q(ζm)Hr is called the field of rationality of G. The field F r(G) canalso be characterized as the field obtained by adjoining to Q the values of the character tableof G. Let

H(C) = {s ∈ (Z/mZ)× : Csi = Ci for all i}

be the stabilizer of C under this action. We define the field of rationality of C to be

F r(C) = Q(ζm)H(C).

Similarly, let

Hwr(C) = {s ∈ (Z/mZ)× : Cs = ϕ(C) for some ϕ ∈ Aut(G)}.

We define the field of weak rationality of C to be Fwr(C) = Q(ζm)Hwr(C). Then

Fwr(C) ⊆ F r(C) ⊆ F r(G).

The group Gal(Q/Fwr(C)) acts on the set of generating tuples g ∈ Σ(C) up to uniformautomorphism, which we denote Σ(C)/Aut(G). For g ∈ Σ(C), the cover f : X = X(g)→ P1has field of moduli M(X, f) equal to the fixed field of the kernel of this action, a numberfield of degree at most dwr = #Σ(C)/Aut(G) over Fwr(C).

Similarly, a G-Galois branched cover f : X → P1 (equipped with its isomorphism i : G ∼−→Aut(X, f)) has field of moduli M(X, f,G) equal to the fixed field of the stabilizer of theaction of Gal(Q/F r(C)), a number field of degree ≤ dr = #Σ(C)/ Inn(G) over F r(C).

22

Therefore, we have the following diagram of fields.

M(X, f,G)

≤dr=#Σ(C)/ Inn(G)M(X, f)

≤dwr=#Σ(C)/Aut(G) F r(C)

Fwr(C)

Q

The simplest case of this setup is as follows. We say that C is rigid if the action of Inn(G)on Σ(C) is transitive. By the above, if Σ(C) is rigid then this action is simply transitive andso endows Σ(C) with the structure of a torsor under G = Inn(G). In this case, the diagramcollapses to

M(X, f,G) = F r(C) ⊇M(X, f) = Fwr(C).More generally, we say that C is weakly rigid if for all g, g′ ∈ Σ(C) there exists ϕ ∈ Aut(G)such that ϕ(g) = g′. (Coombes and Harbater [15] say inner rigid and outer rigid for rigidand weakly rigid, respectively.) If C is weakly rigid, and X = X(g) with g ∈ C, thenM(X, f) = Fwr(C) and the group Gal(M(X, f,G)/F r) injects canonically into the outerautomorphism group Out(G) = Aut(G)/ Inn(G).

We summarize the above discussion in the following proposition.

Proposition 6.1. Let G be a group with trivial center. Let g = (g1, . . . , gn) be a generatingtuple for G and let C = (C1, . . . , Cn), where Ci is the conjugacy class of gi. Let P1, . . . , Pn ∈P1(Q). Then the following statements hold.

(a) There exists a branched covering f : X → P1 with ramification type C = (C1, . . . , Cn)over the points P1, . . . , Pn and an isomorphism G

∼−→ Aut(X, f), all defined over Q.(b) The field of moduli M(X, f) of f is a number field of degree at most

dwr = #Σ(C)/Aut(G)

over Fwr(C).(c) The field of moduli M(X, f,G) of f as a G-Galois branched cover is a number field

of degree at most

dr = #Σ(C)/ Inn(G)

over F r(C).

7. Conjugacy classes, fields of rationality

Let p be a prime number and q = pr a prime power. Let Fq be a field with q elementsand algebraic closure Fq. In this section, we record some basic but crucial facts concerning

23

conjugacy classes and automorphisms in the finite matrix groups arising from GL2(Fq); seeHuppert [33, §II.8] for a reference.

First let g ∈ GL2(Fq). By the Jordan canonical form, exactly one of the following holds:(1) The characteristic polynomial f(g;T ) ∈ Fq[T ] has two repeated roots (in Fq), and

hence g is either a scalar matrix (central in GL2(Fq)) or g is conjugate to a matrix

of the form

(t 10 t

)with t ∈ F×q ; or

(2) f(g;T ) has distinct roots (in Fq) and the conjugacy class of g is uniquely determinedby f(g;T ), and we say g is semisimple.

Let PGL2(Fq) = GL2(Fq)/F×q and let g be the image of g under the natural reduction map

GL2(Fq) → PGL2(Fq). We have g = 1 iff g is a scalar matrix. If g is conjugate to(t 10 t

),

then g is conjugate to

(1 10 1

), and we say that g is unipotent. If f(g;T ) is semisimple, then

in the quotient the conjugacy classes associated to f(g;T ) and f(cg;T ) = c2f(g; c−1T ) forc ∈ F×q become identified. If f(g;T ) factors over Fq then g is conjugate in PGL2(Fq) to the

image of a matrix

(1 00 x

)with x ∈ F×q \ {1}, and we say that g is split (semisimple). The

set of split semisimple conjugacy classes in PGL2(Fq) is therefore in bijection with the set

(7.1) {{x, x−1} : x ∈ F×q \ {1}}.

There are (q−3)/2+1 = (q−1)/2 such classes if q is odd, and (q−2)/2 = q/2−1 such classesif q is even. On the other hand, the conjugacy classes of semisimple elements with irreduciblef(g;T ) are in bijection with the set of monic, irreducible polynomials f(T ) ∈ Fq[T ] of degree2 up to rescaling x 7→ ax with a ∈ F×q . There are q(q−1)/2 such monic irreducible quadraticpolynomials T 2 − aT + b; for any such polynomial with a 6= 0, there is a unique rescalingsuch that a = 1; when q is odd, there is a unique such polynomial with a = 0 up to rescaling.Therefore, the total number of conjugacy classes is (q(q−1)/2)/(q−1) = q/2 when q is evenand (q(q − 1)/2− (q − 1)/2) /(q− 1) + 1 = (q− 1)/2 when q is odd. Equivalently, the set ofnonsplit semisimple conjugacy classes in PGL2(Fq) is in bijection with the set

(7.2) {{y, yq} : y ∈ (Fq2 \ Fq)/F×q }

by taking roots.Now let g ∈ SL2(Fq)\ with g 6= ±1. Suppose first that f(g;T ) has a repeated root,

necessarily ±1; then we say that g is unipotent. For u ∈ Fq, let U(u) =(

1 u0 1

). Using

Jordan canonical form we find that g is conjugate to ±U(u) for some u ∈ F×q . The matricesU(u) and U(v) are conjugate if and only if uv−1 ∈ F×2q . Thus, if q is odd there are fournontrivial conjugacy classes associated to characteristic polynomials with repeated roots,whereas is q is even there is a single such conjugacy class.

Otherwise the element g is semisimple and so g is conjugate in SL2(Fq) to the matrix(0 −11 tr(g)

)by rational canonical form, and the trace map provides a bijection between the

set of conjugacy classes of semisimple elements of SL2(Fq) and elements α ∈ Fq with α 6= ±2.24

Finally, we give the corresponding description in PSL2(Fq) = SL2(Fq)/{±1}. When p =2 we have PSL2(Fq) = SL2(Fq), so assume that p is odd. Then the conjugacy classesof the matrices U(u) and −U(u) in SL2(Fq) become identified in PSL2(Fq), so there areprecisely two nontrivial unipotent conjugacy classes, each consisting of elements of orderp. If g is a semisimple element of SL2(Fq) of order a, then the order of its image ±g inPSL2(Fq) is a/gcd(a, 2). We define the trace of an element ±g ∈ PSL2(Fq) to be tr(±g) ={tr(g),− tr(g)} ⊆ Fq and define the trace field of ±g to be Fp(tr(±g)). The conjugacyclass, and therefore the order, of a semisimple element of PSL2(Fq) is then again uniquelydetermined by its trace. (This is particular to PSL2(Fq)—the trace does not determine aconjugacy class in PGL2(Fq)!)

We now describe outer automorphism groups (see e.g. Suzuki [73]). The p-power Frobeniusmap σ, acting on the entries of a matrix by a 7→ ap, gives an outer automorphism ofPSL2(Fq) and PGL2(Fq), and in fact Out(PGL2(Fq)) = 〈σ〉. When p is odd, the map τgiven by conjugation by an element in PGL2(Fq) \ PSL2(Fq) is also an outer automorphismof PSL2(Fq), and these maps generate Out(PSL2(Fq)):

(7.3) Out(PSL2(Fq)) '

{〈σ, τ〉, if p is odd;〈σ〉, if p = 2.

In particular, the order of Out(PSL2(Fq)) is 2r if p is odd and r if p = 2. From the embeddingPGL2(Fq) ↪→ PSL2(Fq2), given explicitly by ±g 7→ ±(det g)−1/2g, we may also view the outerautomorphism τ as conjugation by an element of PSL2(Fq2) \ PSL2(Fq).

We conclude this section by describing the field of rationality (as defined in section 6) forthese conjugacy classes.

Lemma 7.4. Let g ∈ PGL2(Fq) have order m. Then the field of rationality of the conjugacyclass C of g is

F r(C) =

{Q(λm), if g is semisimple;Q, if g is unipotent;

and the field of weak rationality of C is

Fwr(C) =

{Q(λm)〈Frobp〉, if g is semisimple;Q, if g is unipotent.

Proof. A power of a unipotent conjugacy class is unipotent or trivial so its field of rationalityand weak rationality is Q.

If C is split semisimple, corresponding to {x, x−1} by (7.1) with x ∈ F×q \ {1} then g 7→ gsfor s ∈ (Z/mZ)× corresponds to the map x 7→ xs and it stabilizes the set {x, x−1} (resp. upto an automorphism of PGL2(Fq)) if and only if s ∈ 〈−1〉 ⊆ (Z/mZ)× (resp. s ∈ 〈−1, p〉).

Next consider the case where C is nonsplit semisimple, corresponding to {y, yq} by (7.2)with y ∈ (Fq2 \ Fq)/F×q . Then the map g 7→ gs again with s ∈ (Z/mZ)× corresponds tothe map y 7→ ys. We have y ∈ F×q2 ' Z/(q

2 − 1)Z, with the image of F×q the subgroup(q+ 1)Z/(q2− 1)Z, so the set {y, yq} is stable if and only if s ∈ {1, q} = 〈−1〉 (mod m), andthe set is stable up to an automorphism of PGL2(Fq) if and only if s ∈ 〈−1, p〉 ⊂ Z/mZ, sowe have the same result as in the split case. �

For an odd prime p, we abbreviate p∗ = (−1)(p−1)/2p. Recall that q = pr.25

Lemma 7.5. Let ±g ∈ PSL2(Fq) have order m. Then the field of rationality of the conjugacyclass C of g is

F r(C) =

Q(λm), if g is semisimple;Q(√p∗), if g is unipotent and pr is odd;

Q, otherwise.The field of weak rationality of C is

Fwr(C) =

{Q(λm)〈Frobp〉, if g is semisimple;Q, otherwise,

where Frobp ∈ Gal(Q(λm)/Q) is the Frobenius element associated to the prime p.Proof. First, suppose ±g = ±U(u) is unipotent with u ∈ F×q . Then for all integers s prime top, we have (±g)s = ±U(su). Thus, the subgroup of (Z/pZ)× = F×p stabilizing C is preciselythe set of elements of F×p which are squares in F×q . Thus if p = 2 or r is even, this subgroupis all of F×p so that the field of rationality of C is Q, whereas if pr is odd this subgroupis the unique index two subgroup of F×p and the corresponding field of rationality for C inPSL2(Fq) is Q(

√p∗).

Next we consider semisimple conjugacy classes. By the trace map, these classes are inbijection with ±t ∈ ±Fq \ {±2}. The induced action on the set of traces is given by±t = ±(z + 1/z) 7→ ±(zs + 1/zs) for s ∈ (Z/mZ)× where z is a primitive mth root of unity.From this description, we see that the stabilizer is 〈−1〉 ⊆ (Z/mZ)×.

A similar analysis yields the field of weak rationality. If C is unipotent then τ identifiesthe two unipotent conjugacy classes so the field of weak rationality is always Q. If C issemisimple then σ identifies C with Cp so the stabilizer of ±t is 〈−1, p〉 ⊆ (Z/mZ)×, thefield fixed further under the Frobenius Frobp. �

8. Subgroups of PSL2(Fq) and PGL2(Fq) and weak rigidity

The general theory developed for triples in section 6 can be further applied to the groupsPSL2(Fq) (and consequently PGL2(Fq)) using work of Macbeath [41], which we recall in thissection. See also Langer and Rosenberger [39], who give an exposition of Macbeath’s workin our context.

Let q be a prime power. We begin by considering triples g = (g1, g2, g3) with gi ∈ SL2(Fq)– we remind the reader that the terminology implies g1g2g3 = 1 and does not imply that〈g1, g2, g3〉 = SL2(Fq) – with an eye to understanding the image of the subgroup generatedby g1, g2, g3 in PSL2(Fq) according to the traces of the corresponding elements. Long periodsof consternation have taught us that the difference between a matrix and a matrix up tosign plays an important role here, and so we keep this in our notation. Moreover, becausewe will be considering other kinds of triples, we refer to g = (g1, g2, g3) as a group triple.

A trace triple is a triple t = (t1, t2, t3) ∈ F3q. For a trace triple t, let T (t) denote the set ofgroup triples g such that tr(gi) = ti for i = 1, 2, 3. The group Inn(SL2(Fq)) = PSL2(Fq) actson T (t) by conjugation.

Proposition 8.1 (Macbeath [41, Theorem 1]). For all trace triples t, the set T (t) is nonempty.

To a group triple g = (g1, g2, g3) ∈ SL2(Fq)3, we associate the order triple (a, b, c) by lettinga be the order of ±g1 ∈ PSL2(Fq), and similarly b the order of ±g2 and c the order of ±g3.

26

Without loss of generality, as in the definition of the triangle group (2.2) we may assumethat an order triple (a, b, c) has a ≤ b ≤ c.

Our goal is to give conditions under which we can be assured that a group triple generatesPSL2(Fq) or PGL2(Fq) and not a smaller group. We do this by placing restrictions on theassociated trace triples, which come in three kinds.

A trace triple t is commutative if there exists g ∈ T (t) such that the group ±〈g1, g2, g3〉 ⊆PSL2(Fq) is commutative. By a direct calculation, Macbeath proves that a triple t is com-mutative if and only if the ternary Fq-quadratic form

x2 + y2 + z2 + t1yz + t2xz + t3xy

is singular [41, Corollary 1, p. 21], i.e. if and only if its (half-)discriminant

(8.2) d(t) = d(t1, t2, t3) = t21 + t

22 + t

23 − t1t2t3 − 4

is zero. If a trace triple t is not commutative, then the order triple (a, b, c) is the same forany g ∈ T (t): the trace uniquely defines the order of a semisimple or unipotent element, andif some gi is scalar in g ∈ T (t) then the group it generates is necessarily commutative.

A trace triple t is exceptional if there exists a triple g ∈ T (t) with order triple equal to(2, 2, c) with c ≥ 2 or one of(8.3) (2, 3, 3), (3, 3, 3), (3, 4, 4), (2, 3, 4), (2, 5, 5), (5, 5, 5), (3, 3, 5), (3, 5, 5), (2, 3, 5).

Put another way, a trace triple t is exceptional if there exists g ∈ T (t) whose order triple isthe same as that of a triple of elements of SL2(Fq) that generates a finite spherical trianglegroup in PSL2(Fq).

Finally, a trace triple t is projective if for all g = (g1, g2, g3) ∈ T (t), the subgroup±〈g1, g2, g3〉 ⊆ PSL2(Fq) is conjugate to a subgroup of the form PSL2(k) or PGL2(k) fork ⊆ Fq a subfield.

Remark 8.4. There are no trace triples which are both projective and commutative. Aprojective trace triple may be exceptional, but the possibilities can be explicitly describedas follows. A trace triple is exceptional if there is a homomorphism from a finite sphericalgroup to PSL2(Fq) with order triple as given in (8.3); a trace triple is projective if the imageis conjugate to PSL2(k) or PGL2(k) for k ⊆ Fq a subfield. From the classification of finitespherical groups, this homomorphism must be one of the following exceptional isomorphisms:

D6 ' GL2(F2), A4 ' PSL2(F3), S4 ' PGL2(F3), or A5 ' PGL2(F4) ' PSL2(F5).Any trace triple t with Fp(t) = Fp(t1, t2, t3) = Fq that is both exceptional and projectivecorresponds to one of these isomorphisms.

We now come to Macbeath’s classification of subgroups of PSL2(Fq) generated by twoelements.

Theorem 8.5 ([41, Theorem 4]). Every trace triple t is exceptional, commutative or projec-tive.

Example 8.6. We illustrate the above with the case q = 7. There are a total of 73 = 243trace triples.

First, the trace triples where the order triple is not well-defined are the trace triples

(2, 2, 2), (2,−2,−2), (−2, 2,−2), (−2,−2, 2)27

which are commutative. The role of multiplication by −1 plays an obvious role here, so inthis example, for a trace triple t = (t1, t2, t3) we say that a trace triple agrees with t with aneven number of signs if it is one of

(t1, t2, t3), (t1,−t2,−t3), (−t1, t2,−t3), (−t1,−t2, t3).

Put this way, the trace triples where the order triple is not well-defined are those agreeingwith (2, 2, 2) with an even number of signs. We define odd number of signs analogously. Foreach of these four trace triples, there exists a group triple g with order triple (1, 1, 1), (1, 7, 7),or (7, 7, 7).

The other commutative trace triples are:

(2, 0, 0), with any signs having orders (1, 2, 2)

(2, 1, 1), with even number of signs having orders (1, 3, 3)




(1, 1,−1), with odd number of signs having orders (3, 3, 3)

Indeed, these are all values (t1, t2, t3) ∈ F3q such that

d(t1, t2, t3) = t21 + t

22 + t

23 − t1t2t3 − 4 = 0.

The three commutative trace triples (0, 0, 0), (0, 3, 3), (1, 1,−1) are also exceptional. Theremaining exceptional triples are:







28

All other triples are projective:









(3, 3,−2), with odd number of signs having orders (4, 4, 7)(3, 2, 2), with any signs having orders (4, 7, 7)

(2, 2,−2), with odd number of signs having orders (7, 7, 7)We note that the triples (1, 3,−3) with an odd number of signs in fact generate PSL2(F7)—

but the triple is not projective. In particular, we observe that one cannot deduce that anonsingular trace triple is projective by looking only at its order triple.

Finally, the issue that this example is supposed to make clear is that changing signson a trace triple may change the subgroup of PSL2(Fq) that a corresponding group triplegenerates. Indeed, changing an odd number of signs on a group triple does not yield a grouptriple! We address the parity of signs in the next lemma.

The role of −1 and the parity of these signs (taking an even or odd number) is a key issuethat will arise and so we address it now.

Lemma 8.7. Let t = (t1, t2, t3) ∈ F3q be a trace triple.(a) There are bijections

T (t)↔ T (t1,−t2,−t3)↔ T (−t1, t2,−t3)↔ T (−t1,−t2, t3)(g1, g2, g3) 7→ (g1,−g2,−g3) 7→ (−g1, g2,−g3) 7→ (−g1,−g2, g3)

which preserve the subgroups generated by each triple. In particular, if t is commu-tative (resp. exceptional, projective), then so is each of

(t1,−t2,−t3), (−t1, t2,−t3), (−t1,−t2, t3).(b) Suppose q is odd. If t is commutative, then (−t1, t2, t3) is commutative if and only if

t1t2t3 = 0.

Proof. Part (a) is clear. As for part (b): the trace triple t is commutative if and only ifd(t1, t2, t3) = 0. So (−t1, t2, t3) is also commutative if and only if d(−t1, t2, t3) = 0 if andonly if d(t1, t2, t3)− d(−t1, t2, t3) = 2t1t2t3 = 0, as claimed. �

Let `/k be a separable quadratic extension. We say that t ∈ ` is a squareroot from kif t = 0 or t =

√u with u ∈ k× \ k×2. A trace triple t is irregular [41, p. 28] if the field

Fp(t) = Fp(t1, t2, t3) ⊆ Fq has a subfield k ⊆ Fp(t) such that(i) [Fp(t) : k] = 2 and(ii) after reordering, we have t1 ∈ k and t2, t3 are squareroots from k.

29

Otherwise, t is regular. Of course, if [Fp(t) : Fp] is odd—e.g., if [Fq : Fp] is odd—then t isnecessarily regular.

Proposition 8.8 ([41, Theorem 3]). Let g generate a projective subgroup G = ±〈g1, g2, g3〉 ⊆PSL2(Fq) and let t be its trace triple.

(a) Suppose t is regular. Then G is conjugate in PSL2(Fq) to PSL2(Fp(t)).(b) Suppose t is irregular, and let k0 be the unique index 2 subfield of Fp(t). Then G is

conjugate in PSL2(Fq) to either PSL2(Fp(t)) or PGL2(k0).(c) Suppose k = Fq. Then the number of orbits of Inn(SL2(Fq)) = PSL2(Fq) on T (t) is

2 or 1 according as p is odd or p = 2.(d) For all g′ ∈ T (t), there exists m ∈ SL2(Fq) such that m−1gm = g′.

We say that a trace triple t is of PSL2-type (resp. of PGL2-type) if t is projective and forall g ∈ T (t) the group ±〈g1, g2, g3〉 is conjugate to PSL2(k) (resp. PGL2(k0)); by Proposition8.8(a), every projective triple is either of PSL2-type or of PGL2-type.

We now transfer these results to trace triples in the projective groups PSL2(Fq). The pas-sage from SL2(Fq) to PSL2(Fq) identifies conjugacy classes whose traces have opposite signs,so associated to a triple of conjugacy classes C in PSL2(Fq) is a trace triple (±t1,±t2,±t3),which we abbreviate ±t (remembering that the signs may be taken independently). We call±t a trace triple up to signs.

Let ±t be a trace triple up to signs. We say ±t is commutative if there exists ±g ∈ T (±t)such that ±〈g1, g2, g3〉 is commutative. We say ±t is exceptional if there exists a lift of ±t toa trace triple t such that the associated order triple (a, b, c) is exceptional. Finally, we say±t is projective if all lifts t of ±t are projective, and partly projective if there exists a lift t of±t that is projective.

Lemma 8.9. Every trace triple up to signs is exceptional, commutative, or partly projective.

Proof. This follows from Theorem 8.5 and Lemma 8.7. �

To a nonsingular trace triple up to signs ±t, we associate the order triple (a, b, c) as theorder triple associated to any lift t of ±t = (±t1,±t2,±t3); this is well defined because wetook orders of elements in PSL2(Fq) from the very beginning. We assume that a ≤ b ≤ c;Remark 1.2 explains why this is no loss of generality.

For a triple of conjugacy classes C = (C1, C2, C3) of PSL2(Fq), recall we have defined Σ(C)to be the set of generating triples g = (g1, g2, g3) such that gi ∈ Ci.

Proposition 8.10. Let C be a triple of conjugacy classes in PSL2(Fq). Let ±t be theassociated trace triple up to signs, let Fq = Fp(±t), and let (a, b, c) the associated ordertriple. Suppose that ±t is partly projective and not exceptional, and let G = ±〈g1, g2, g3〉 ⊆PSL2(Fq).

Then the values #Σ(C)/ Inn(G) and #Σ(C)/Aut(G) are given in the following table:30

p a abc G #Σ(C)/ Inn(G) #Σ(C)/Aut(G)

p = 2 − − − 1 1

p > 2 a = 2p | abc − 1 1p - abc PGL2 1 1p - abc PSL2 2 1

p > 2 a 6= 2p | abc − ≤ 2 ≤ 2p - abc PGL2 ≤ 2 ≤ 2p - abc PSL2 ≤ 4 ≤ 2

Proof. Suppose p = 2. Then PSL2(Fq) = SL2(Fq) and by Proposition 8.8(b) the triple C isrigid, i.e. we have #Σ(G)/ Inn(G) = #Σ(G)/Aut(G) = 1. This gives the first row of thetable. So from now on we suppose p > 2.

Let t = (t1, t2, t3) ∈ F3q be a lift of ±t. Let ±g,±g′ ∈ Σ(C); lift them to g, g′ in SL2(Fq)3such that g1g2g3 = ±1 and g′1g′2g′3 = ±1 and such that tr(gi) = tr(g′i) = ti.

Case 1: Suppose a = 2. Then t1 = tr(g1) = 0 = − tr(g1), so changing the signs of g1 andg′1 if necessary, we may assume that g, g

′ are triples (that is, g1g2g3 = g′1g′2g′3 = 1). Then

by Proposition 8.8(c), there exists m ∈ SL2(Fq) such that m conjugates g to g′. Since theelements of ±g generate G by hypothesis and the elements of ±g′ lie in G, it follows thatconjugation by m induces an automorphism ϕ of G, so ϕ(g) = g′, and C is weakly rigid.This gives the entries in the last column for p > 2 and a = 2.

Case 1(a): Suppose a = 2 and p | abc. Then at least one conjugacy class is unipotent andso the two orbits of the set T (t) under Out(PSL2(Fq)) correspond to two different conjugacyclass triples and only one belongs to C. Therefore the triple is in fact rigid.

Case 1(b): Suppose a = 2 and p - abc. First, suppose that G is of PGL2-type. ThenG ' PGL2(F√q) and Out(PGL2(F√q)) = 〈σ〉; and since Fq = Fp(t), the stabilizer of 〈σ〉acting on t is trivial as in the analysis following (7.3), and hence the orbits must be alreadyidentified by conjugation in PGL2(Fq), so the triple is in fact rigid.

Second, suppose that G is of PSL2-type. Then since Fq = Fp(±t) we have G = PSL2(Fq).Because p - abc, all conjugacy classes Ci ∈ C are semisimple and so are preserved underautomorphism. From Proposition 8.8(b), we see that there are two orbits of PSL2(Fq) actingby conjugation on Σ(C) and the element τ ∈ Out(PSL2(Fq)) induced by conjugation by anelement of PGL2(Fq) \ PSL2(Fq) identifies these orbits: they are identifed by some elementof Out(PSL2(Fq)), but since Fq = Fp(t) the stabilizer of 〈σ〉 acting on t is again trivial.

This completes Case 1 and the table for p > 2 and a = 2. Note that in this case, thechoice of the lift t does not figure in the analysis.

Case 2: Suppose a > 2. Now either g′ is already a triple, or changing the sign of g1 wehave g′ is a triple with trace triple t′ = (−t1, t2, t3). By Lemma 8.7, this is without loss ofgenerality.

Case 2(a): Suppose t′ = t. Then the same analysis as in Case 1 shows that g′ is obtainedfrom g by an automorphism ϕ of G, and this automorphism can be taken to be inner exceptwhen p - abc and G = PSL2(Fq), in which case up to conjugation there are two triples.

Case 2(b): Suppose t′ 6= t. Then clearly g′ is not obtained from g by an inner automor-phism. If g′ = ϕ(g) with ϕ an outer automorphism, then after conjugation in SL2(Fq), as inCase 2(a), we may assume that ϕ = σj is a power of the p-power Frobenius automorphism

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σ, with σj(t1) = −t1 and σj(t2) = t2 and σj(t3) = t3 (with slight abuse of notation). Butagain Fq is generated by the trace triple, so the fixed field k of σj contains t2, t3 and Fq is aquadratic extension of k, generated by t1, and t1 is a squareroot from k.

This concludes Case 2, with the stated inequalities: we have at most twice as many triplesas in Cases 1(a) and 1(b).

We have equality in the first column if and only if t′ is projective: otherwise, t′ is com-mutative, as in the proof of Proposition 8.10, and all g′ ∈ t′ generate affine or commutativesubgroups of PSL2(Fq) since they are singular [41, Theorem 2], and any such triple does notbelong to Σ(C)—the trace triple up to signs is not exceptional so any group generated by acorresponding triple is not projective. In the second column, we have equality if and only ift′ is projective and we are not in the special case described in 2(b). �

9. Proof of Theorems

In this section, we give proofs of the main theorems A, B, and C.We begin with Theorem A, which follows from the following theorem.

Theorem 9.1. Let (a, b, c) be a hyperbolic triple with a, b, c ∈ Z≥2 ∪ {∞}. Let p be a primeof E(a, b, c) with residue field Fp lying above the rational prime p, and suppose p - abc. Ifp | 2, suppose further that (a, b, c) is not of the form (mk,m(k+1),mk(k+1)) with k,m ∈ Z.Let a] = p if a =∞ and a] = a otherwise, and similarly with b, c.

Then there exists a G-Galois Bely̆ı map

X(a, b, c; p)→ P1

with ramification indices (a], b], c]), where

G =

{PSL2(Fp), if p splits completely in F (a, b, c);PGL2(Fp), otherwise.

Proof. Let (a, b, c) be a hyperbolic triple with a, b, c ∈ Z≥2 ∪ {∞}. Let p be a prime of thefield

E(a, b, c) = Q(λa, λb, λc, λ2aλ2bλ2c)and let P be a prime of

F (a, b, c) = Q(λ2a, λ2b, λ2c)above p above the rational prime p - abc.

Then by Proposition 5.22, we have a homomorphism

φ : ∆(a, b, c)→ PSL2(FP)with trφ(δ̄s) ≡ ±λ2s (mod P) for s = a, b, c whose image lies in the subgroup PSL2(FP) ∩PGL2(Fp). We have [FP : Fp] ≤ 2 and

(9.2) PSL2(FP) ∩ PGL2(Fp) =

{PSL2(Fp), if FP = Fp;PGL2(Fp), if [FP : Fp] = 2.

Let ∆(a, b, c; p) be the kernel of the homomorphism φ : ∆(a, b, c) → PSL2(FP). Thegenerators δ̄s of ∆ (for s = a, b, c) give rise to a triple g = (g1, g2, g3), namely g1 = φ(δ̄a),

g2 = φ(δ̄b), g3 = φ(δ̄c), with trace triple up to signs

±t = (±t1,±t2,±t3) ≡ (±λ2a,±λ2b,±λ2c) (mod P).32

The map of complex algebraic curves

f : X = X(a, b, c; p) = ∆(a, b, c; p)\H → ∆(a, b, c)\H ' P1

is a G-Galois Bely̆ı map by construction, where G = ∆(a, b, c)/∆(a, b, c; p). It is our task tospecify G by our understanding of subgroups of PSL2(FP).

First, we dispose of the exceptional triples. Since (a, b, c) is hyperbolic, this leaves the fivetriples

(a], b], c]) = (3, 4, 4), (2, 5, 5), (5, 5, 5), (3, 3, 5), (3, 5, 5).

Each of these triples is arithmetic, by work of Takeuchi [77], and the result follows fromwell-known properties of Shimura curves—something that could be made quite explicit ineach case, if desired.

Second, we claim that the triple ±t is not commutative. From (8.2), the triple t iscommutative if and only if

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